Let \(v_i\in S_pN\)  for \(i \in \mathbf {N}\)  converge to \(v\) .
We set
\(C={\tau _{\text{con}}}(p,v), \quad B=\liminf _{i \rightarrow \infty }{\tau _{\text{con}}}(p,v_i), \quad \hbox{ and } \quad A=\limsup _{i\rightarrow \infty }{\tau _{\text{con}}}(p,v_i).\) 
It suffices to show that \(A\le C \le B\) .
We assume first that \(C=\infty \) . If \(A<\infty \)  we choose a sub-sequence \(v_{i_k}\)  of \(v_i\)  such that \({\tau _{\text{con}}}(p,v_{i_k})\)  converges to \(A\) . Then
\(\det (\mathrm {D}\exp _{p}({\tau _{\text{con}}}(p,v_{i_k})v_{i_k}))=0,\) 
and the smoothness of the exponential map gives \(\det (\mathrm {D}\exp _{p}(Av))=0\)  yielding that \(\exp _{p}(Av)\)  is conjugate to \(p\)  along \(\gamma _{p,v}\) . This implies that \({\tau _{\text{con}}}(p,v)\le A\) , which is impossible. By the same argument we see that \(B=\infty \) .
Let \(C<\infty \) , and by the same limiting argument as above we get \(C\le B\) .
Then we show that \(A\le C\) . Choose a sub-sequence \(v_{i_k}\)  such that \({\tau _{\text{con}}}(p,v_{i_k})\rightarrow A\)  as \(k\rightarrow \infty \) . For a sake of contradiction we suppose that \(A>C\) . By the definition of the conjugate distance function we have that \(q=\gamma _{p,v}(C)\)  is the first conjugate to \(p\)  along \(\gamma _{p,v}\) . By [1]} for any \(\varepsilon \in (0,A-C)\)  there exists a piecewise smooth vector field \(X\)  on the geodesic segment \(\gamma _{p,v}\colon {[0,C+\varepsilon ]} \rightarrow N\) , that vanishes on 0 and \(C+\varepsilon \)  such that the index form of \(\gamma _{p,v}\)  over \(X\)  is strictly negative. That is
\(I_{v}(X,X):=\int _{0}^{C+\varepsilon }\langle \mathrm {D}_t X, \mathrm {D}_t X\rangle _g+\langle R(\dot{\gamma }_{p,v},X)\dot{\gamma }_{p,v}, X \rangle _g \; \mathrm {d}t<0.\) 
Here we used the notation \(\mathrm {D}_t\)  for the covariant differentiation along \(\gamma _{p,v}\) . The capital \(R\)  stands for the Riemannian curvature tensor.
We choose vectors \(E_1,\ldots ,E_n\)  of \(T_pN\)  that form a basis of \(T_pN\)  and extend them on \(\gamma _{p,v}(t)\)  for \(t \in [0,C+\varepsilon ]\)  via the parallel transport. Since parallel transport is an isomorphism the vector fields \(\lbrace E_1(t), \ldots ,E_{n}(t)\rbrace \)  constitute a basis of \(T_{ \gamma _{p,v}(t)}M\) . We write \(X(t)=X^j(t)E_j(t)\) . Since \(X\)  is piecewise smooth it holds that the component functions \(X^j(t)\)  are piecewise smooth. This lets us to `extend' \(X\)  on \(\gamma _{p,v_{i_k}}\)  by the formula
\(X_k(t)=X^j(t)E_j^k(t),\) 
where the vector field \(E_j^k(t)\)  is the parallel transport of \(E_j\)  along \(\gamma _{p,v_{i_k}}\) . Thus \(X_k\)  is a piecewise smooth vector field on \(\gamma _{p,v_{i_k}}\)  that vanishes at \(t=0\)  and \(t=C+\varepsilon \) .
Since
\(v_{i_k}\rightarrow v, \hbox{ as $k \rightarrow \infty $},\) 
it holds that
\(\gamma _{p,v_{i_k}}(t) \rightarrow \gamma _{p,v}(t), \quad \hbox{ and } \quad E^k_j(t)\rightarrow E_j(t), \quad \hbox{uniformly in $t \in [0,C+\varepsilon ]$ as $k \rightarrow \infty $. }\) 
Therefore by (REF ), (REF ), and the continuity of the Levi-Civita connection and the Riemannian curvature tensors we have
\(I_{v_{i_k}}(X_k, X_k)<0, \quad \hbox{ for large enough $k\in \mathbf {N}$.}\) 
By [1]} there exists \(s_k \in (0,C+\varepsilon ]\)  so that \(\gamma _{p,v_{i_k}}(0)\)  and \(\gamma _{p,v_{i_k}}(s_k)\)  are conjugate points. Therefore we must have \(s_k\ge {\tau _{\text{con}}}(p,v_{i_k})\)  and we arrive in a contradiction
\({\tau _{\text{con}}}(p,v_{i_k}) < A.\) 
This ends the proof.
